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NUMBER FIELD ZETA INTEGRALS and L-FUNCTIONS the Basic Zeta

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NUMBER FIELD ZETA INTEGRALS and L-FUNCTIONS the Basic Zeta NUMBER FIELD ZETA INTEGRALS AND L-FUNCTIONS The basic zeta function is X ζ(s) = n−s; Re(s) > 1: n≥1 Riemann proved that the completed basic zeta function, Z(s) = π−s=2Γ(s=2)ζ(s); Re(s) > 1; has an entire meromorphic continuation that satisfies the functional equation Z(1 − s) = Z(s); s 2 C: One of his proofs proceeds by arguing as follows. • Poisson summation shows that the theta function, X 2 θ(t) = e−πin t; t > 0; n2Z is a modular form, θ(1=t) = t1=2θ(t): • Z is the Mellin transform (essentially a Fourier transform) of θ, Z 1 s=2 dt Z(s) = 2 (θ(t) − 1)t ; Re(s) > 1: t>0 t As t ! 1 the integrand decays rapidly regardless of the value of s, posing no obstacle to the integral's convergence at its improper upper limit of integration. But as t ! 0+ the integrand is O(t(Re(s)−1)=2 dt=t), requiring Re(s) > 1 for the integral to converge at its improper lower limit. • The transformation law for θ shows that in fact Z 1 s=2 (1−s)=2 dt 1 1 Z(s) = 2 (θ(t) − 1)(t + t ) − − : t>1 t s 1 − s With the problematic lower limit of integration no longer present, the right side is now sensible as a meromorphic function for all s 2 C, and it visibly satisfies the functional equation. Similar methods apply to any Dirichlet L-function, X L(s; χ) = χ(n)n−s; Re(s) > 1: n≥1 The Dirichlet L-function has a suitable completion Λ(s; χ); Re(s) > 1; involving a factor similar to the factor π−s=2Γ(s=2) for the basic zeta function, but now taking into account the parity and the conductor of χ. Again the completion has a continuation and a functional equation. The calculation now involves a Gauss sum along with the other ingredients, and the functional equation includes a factor called a root number, W (χ)Λ(1 − s; χ) = Λ(s; χ): 1 2 NUMBER FIELD ZETA INTEGRALS AND L-FUNCTIONS If the Dirichlet character is quadratic then the root number W (χ) is 1. Both the basic zeta function and the Dirichlet L-function are defined over the basic number field Q. The technique of completion, continuation, and functional equation apply more generally to the Dedekind zeta function of a number field k, X −s ζk(s) = Na ; a and even for any L-function associated to a number field k and a so-called Hecke character of k, X −s Lk(s; χ) = χ(a)Na : a But the definition of a Hecke character is complicated, and the calculations and the root number become ever more elaborate as the environment grows. As Artin, Iwasawa, Tate, and perhaps others knew around 1950, working in the environment of the adeles simplifies the calculation decisively. In the adelic environment, the definition of a Hecke character is simple and natural, and the root number emerges naturally from local calculations. After defining the adelic zeta integral and looking at some of its local factors, these notes establish a global adelic version of its continuation and functional equa- tion. Strikingly, the procedure is essentially identical to Riemann's original ar- gument. The exposition of this material is based on a presentation and written materials by Paul Garrett. Then the notes continue to discuss the local factors of the zeta integral, following Kudla's lecture in the Introduction to the Langlands Program volume. Contents Part 1. The Zeta Integral 2 1. Definition of the Zeta Integral 2 2. Local Zeta Integrals 3 2.1. Unramified nonarchimedean places 3 2.2. Real archimedean places 4 2.3. Complex archimedean places 4 Part 2. Global Theory 5 3. Convergence 5 4. Elements of Adelic Harmonic Analysis 6 5. Continuation and Functional Equation 8 Part 3. Local Theory 10 6. The Local Zeta Integral 10 7. Spaces of Distributions 11 7.1. The nonarchimedean unramified case 13 7.2. The nonarchimedean ramified case 14 7.3. The nonarchimedean trivial case 15 8. The Local Functional Equation 16 8.1. Additive characters 16 8.2. The local functional equation 18 NUMBER FIELD ZETA INTEGRALS AND L-FUNCTIONS 3 8.3. Pre-normalizing the "-factor calculation 18 8.4. The "-factor calculation 19 9. The Global "-Factor 21 Part 4. The Final Calculation 22 Part 5. Application: Quadratic "-Factors are Trivial 22 Part 1. The Zeta Integral 1. Definition of the Zeta Integral Let k be a number field, and let A be its adele ring. Recall that k is discrete in A and the quotient A=k is compact. Let J be the idele group of k. Recall that k× is discrete in the unit idele group J1 and the quotient J1=k× is compact. (The statement here is Fujisaki's Lemma, encompassing both the structure theorem for the units group of the integers of k and finiteness of the class number of k. The adelic repackaging of those results as Fujisaki's Lemma is optimal for our purposes here.) Let s be a complex parameter, let × χ : J −! C be a Hecke character of J, and let ' : A −! C be an adelic Schwartz function. Let j j denote the idele norm, and let d× denote Haar measure on J. Then the adelic zeta integral associated to χ and ' is Z Z(s; χ, ') = jαjsχ(α)'(α) d×α: J Here s and χ combine to encode the overall character χ j · js being integrated against the Schwartz function. Since χ need not be discretely parameterized, the parame- ters of the zeta integral thus incorporate some redundancy: Z(s0 + s; χ, ') = Z(s0; χ j · js;') for all s0; s; χ. The integral converges for values of s in a right half-plane. 2. Local Zeta Integrals The Hecke character in the zeta integral decomposes as a product of local char- acters, O × χ = χv : J −! C : v The assertion that the Schwartz function in the zeta integral decomposes similarly, O ' = 'v; each 'v is Schwartz; v is not quite true, since in fact ' is a finite sum of such products, but we treat ' as a monomial from now on without further comment. For a nonarchimedean place v, 4 NUMBER FIELD ZETA INTEGRALS AND L-FUNCTIONS the local Schwartz space consists of the compactly supported locally constant func- tions on kv, and for an archimedean place v the local Schwartz space consists of the smooth functions on kv all of whose derivatives are rapidly decreasing. In consequence of χ and ' factoring, the global zeta integral Z Z(s; χ, ') = jαjsχ(α)'(α) d×α J immediately takes the form of an Euler product of local zeta integrals, Y Z(s; χ, ') = Zv(s; χv;'v) v where for each place v, Z s × Zv(s; χv;'v) = jαjvχv(α)'v(α) dv α: × kv The Euler factorization will play no role in the global continuation and functional equation argument. However, we now examine some local Euler factors. 2.1. Unramified nonarchimedean places. Let v be a finite place of k, and let Ov be the ring of integers of kv. With v fixed, freely drop it from the notation. Let the local character have no discretely-parameterized part, × × s χ : k −! C ; χ(α) = jαj o (for any global Hecke character, this holds at almost all the finite places), and let the local Schwartz function be the characteristic function of the local integers, ' : k −! C;'(α) = 1O(α): To compute the local integral Z Z(s) = jαjsχ(α)'(α) d×α; k× take a uniformizer (valuation-1 element) $. Thus j$j = Np−1 where p is the prime ideal of Ok corresponding to the place v. Normalize the multiplicative Haar measure so that O× has measure 1. Then the local integral is, allowing a slight abuse of notation at the last step, 1 Z Z X Z(s) = jαηjs+so '(αη) d×η d×α = j$`js+so = (1 − χ(p)Np−s)−1: × × × k =O O `=0 Thus the local zeta integral gives an Euler factor of the Hecke L-function at almost all places. And the local Schwartz function is its own Fourier transform, so that at such places there is no need to replace it by its Fourier transform in the local factor of the functional equation. The case of a ramified local character in the nonarchimedean case will be dis- cussed later in the writeup. NUMBER FIELD ZETA INTEGRALS AND L-FUNCTIONS 5 2.2. Real archimedean places. Let kv = R. Consider the trivial character and the Gaussian Schwartz function × × χ : R −! C ; χ(α) = 1; −πα2 ' : R −! C;'(α) = e : The local integral is Z Z 1 s × s −πt2 dt − s s Z(s) = jαj '(α) d α = 2 t e = π 2 Γ : × t 2 R 0 This is the familiar archimedean factor from completing the basic zeta function or an even Dirichlet L-function. On the other hand, consider the sign character (which arises as the infinite part of an odd Dirichlet character viewed as a classical Hecke character, for example) and the simplest odd Schwartz function that incorporates the Gaussian, × × χ : R −! C ; χ(α) = sgn(α); −πα2 ' : R −! C;'(α) = αe : Then the local integral is Z Z 1 s × s+1 −πt2 dt − s+1 s + 1 Z(s) = jαj χ(α)'(α) d α = 2 t e = π 2 Γ : × t 2 R 0 This is the archimedean factor from completing an odd Dirichlet L-function.
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